Convex-envelope construction

But what the pressure p(o), diemical potential M(p), etc., in the constrained system are, depend on the distance L that defines the constraint. If L is very large, the flrrduations within can almost amount to phase separation foe van der Waak loops in p(v) and M(p) would then enclose only small areas, and foe analytic functions p(o), ip(p), etc., would be dose to foe non-analytic functions obtained from them by the equal-areas, double-tangent, or convex-envelope constructions. Tire effect of the constraint with such large L is minimal and in the limit in which L is macroscopic foe thermodynamic properties become those of foe unconstrained fluid. But when L is small, the deviation of p(t>) from the equilibrium pressure in foe unconstrained system at that temperature is considerable, and similarly for foe other thermodynamic functions. 

Buckingham. M. J. in Phase Traiaitioia and Critical Phenomena (ed. C. Domb and M. S. Green) Vol. 2, Chapter 1, Academic Press, London (1972). The convex-envelope construction was used first by Gibbs, J. W. TVans. Cbm. Acad. 2, 382 (1873), reprinted in Collected Works, Vol. 1, p. 33. 

Construction of convex envelopes consists of a system of linear inequalities (it is a typical problem in linear programming see, for example, ref. 31). In the simplest cases a convex envelope can also be constructed directly. This envelope can also be described parametrically without using inequalities. For example, for a system of x1 x2,.. ., xq points, their convex envelope consists of linear combinations l xy +. .. + Xqxq where Xu. . Xq are non-negative values whose sum equals unity... 

The results of the analysis for a system of three isomers for various E are represented in Fig. 9(a)-(b). Here, a convex envelope for the finite multitude (106) is vertically hatched and its union with the multitude G(N) e( ) is horizontally hatched. The whole of the hatched multitude is co-invariant and the unhatched region is just V(E). This example of only four multitudes makes it possible to construct the "unattainability regions that would not be a union of those for submultitudes. Three multitudes each contain one vertex and a fourth [Fig. 9(d)] includes two vertices, corresponding to the cases when the entire mass is concentrated either in Aj or in A2. 

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